Introduction

As of 06-04-2020, more than 1,133,000 cases of COVID-19 have been reported across the world, with more than 62,000 deaths (1). This weekly report presents forecasts of the reported number of deaths in the week ahead and analysis of case reporting trends (case ascertainment) for 42 countries with active transmission.

The accuracy of these forecasts vary with the quality of surveillance and reporting in each country. We use the reported number of deaths due to COVID-19 to make these short-term forecasts as these are likely more reliable and stable over time than reported cases. In countries with poor reporting of deaths, these forecasts will likely represent an under-estimate while the forecasts for countries with few deaths might be unreliable. Our estimates of transmissibility reflect the epidemiological situation at the time of the infection of COVID-19 fatalities. Therefore, the impact of controls on estimated transmissibility will be quantifiable with a delay between transmission and death.

Executive Summary

Forecasts and Transmissibility Estimates

Based on our best estimates of transmissibility, the COVID-19 epidemic is:

stabilising¹ in 14 countries
(Argentina, Austria, Dominican Republic, Greece, Indonesia, Iran, Iraq, Italy, Luxembourg, Malaysia, Morocco, Norway, Panama, South Korea);
growing slowly² in 2 countries (\(1 < R_t < 2\))
(Netherlands, Spain);
growing rapidly³ in 26 countries (\(R_t > 2\))
(Algeria, Belgium, Brazil, Canada, China, Czech Republic, Denmark, Ecuador, Egypt, France, Germany, India, Ireland, Israel, Japan, Mexico, Peru, Philippines, Poland, Portugal, Romania, Sweden, Switzerland, Turkey, United Kingdom, United States of America).

Based on the central trends in the forecasts, the total number of reported deaths in the coming week is expected to be:

relatively small in 14 countries (<100 deaths)
(Argentina, China, Czech Republic, Dominican Republic, Egypt, Greece, Iraq, Japan, Luxembourg, Malaysia, Morocco, Norway, Panama, South Korea);
relatively large in 15 countries (100 to 1000 deaths)
(Austria, Canada, Denmark, Ecuador, India, Indonesia, Ireland, Israel, Mexico, Peru, Philippines, Poland, Portugal, Romania, Sweden),
large in 8 countries (1,000 to 5,000 deaths)
(Algeria, Belgium, Brazil, Germany, Iran, Netherlands, Switzerland, Turkey),
very large in 5 countries (> 5,000 deaths)
(France, Italy, Spain, United Kingdom, United States of America).

if the 95% Credible Interval (CrI) of \(R_t\) spans 1
based on the lower and upper limits respectively of the 95% CrI of \(R_t\)
based on the upper limit of the 95% CrI of \(R_t\)

Detailed Summary

Objectives and caveats

The main objective in this report is to produce forecasts of the number of deaths in the week ahead for each country with active transmission.

We define a country as having active transmission if at least ten deaths were observed in the country in each of the past two weeks. In the analysis for week beginning 29-03-2020, 22 countries/regions were included in the analysis. For the week beginning 05-04-2020, the number of countries/regions included based on these thresholds is 42.
We forecast the number of potential deaths as the reporting of deaths is likely to be more reliable and stable over time than the reporting of cases.
As we are forecasting deaths, the latest estimates of transmissibility reflect the epidemiological situation at the time of the infection of COVID-19 fatalities. Therefore, the impact of controls on estimated transmissibility will be quantifiable with a delay between transmission and death.

A secondary objective of this report is to analyse case ascertainment per country. As well as forecasting ahead, we use the number of reported deaths and of cases reported with a delay (delay from reporting to deaths, see Case Ascertainment method) to analyse the reporting trends per country. If the reporting of cases and deaths were perfect, and the delay between reporting and death is known, the ratio of deaths to delayed cases would equal the Case Fatality Ratio (CFR).

In this analysis, key assumptions are:

The mean underlying CFR is 1.38% (95% Credible Interval (1.23 - 1.53)) (2),
The delay from a case being reported to death follows a gamma distribution with mean 10 days, and standard deviation 2 days.
All deaths due to COVID-19 have been reported in each country.

Key results on transmissibility and forecasting

Key results below are based on an ensemble forecast of two models.

Transmissibility is characterised by the reproduction number \(R_t\), i.e. the average number of cases that one infected individual is likely to infect. Analysis of transmissibility indicates that the reproduction numbers last week (week starting 05-04-2020) were highest in:

Algeria with an estimated median \(R_t\) of 6.88 (95% CrI - 4.06 - 9.96), and
Canada with an estimated median \(R_t\) of 3.85 (95% CrI 2.97 - 5.19).

and were lowest in:

Iraq with an estimated median \(R_t\) of 0.75 (95% CrI 0.22 - 2.11), and
Luxembourg with an estimated median \(R_t\) of 0.53 (95% CrI 0.12 - 1.44).

Forecasts of predicted deaths in the coming week (week starting 05-04-2020) are highest in:

France with an estimated median 23,800 deaths (95% CrI 9,140 - 43,300), and
the United States of America with an estimated median 19,500 deaths (95% CrI 14,700 - 24,500).

and are lowest in:

Iraq with an estimated median 17 deaths (95% CrI 2 - 48), and
Luxembourg with an estimated median 10 deaths (95% CrI 1 - 26).

Forecasts in previous weeks performed well, with 67.7% of the observed daily number of deaths across all countries included in the 95% CrI of the forecast intervals.

Key results on case ascertainment

Case ascertainment was estimated based on the deaths in the previous 2 weeks and reported cases in the 10 days prior to that period. Estimates of case ascertainment were highly variable and, due to the underlying assumption of perfect reporting, are likely to be an underestimate. In particular, community deaths due to COVID-19 are likely under-reported (3). Results indicate that, assuming perfect reporting of deaths, the countries with the highest case ascertainment were:

Luxembourg with an estimated median ascertainment of 100.0% (95% confidence interval (CI) 63.1% - 100.0%), and
Israel with an estimated median ascertainment of 72.8% (95% CI 52.0% - 100.0%).

and again assuming perfect reporting of deaths, the countries with the lowest case ascertainment were:

United Kingdom with an estimated median ascertainment of 3.2% (95% CI 2.8%- 3.6%), and
Algeria with an estimated median ascertainment of 2.5% (95% CI 2.1% - 3.0%).

Based on the estimated ascertainment, we estimated the true size of the epidemic in each country in the previous 7 days (week starting 22-03-2020). Countries with the largest true epidemic size in this period were:

the United States of America with an estimated median size of 1,590,000 (95% CI 1,420,000 - 1,770,000), and
Spain with an estimated median size of 978,000 (95% CI 875,000 - 1,090,000).

and countries with the lowest true epidemic size in this period were:

Norway with an estimated median size of of 2,340 (95% CI 1,980 - 2,820), and
Luxembourg with an estimated median size of of 1,680 (95% CI 1,340 - 2,100).

Methods

We define a country to have active transmission if at least ten deaths were observed in the country in the last two consecutive weeks. We intend to produce forecasts every week, for the week ahead. Ensemble forecasts are produced from the outputs of three different models.

Our main analysis assumes a gamma distributed serial interval with mean 6.48 days and standard deviation of 3.83 days following (4). The serial interval estimates observed from various studies thus far may be biased toward lower values due to observation bias whereby, in contact tracing studies, long serial intervals tend to be under-represented. To account for this, as a sensitivity analysis, we also use a shorter serial interval of mean 4.80 days and standard deviation of 2.70 days (5). Results using this shorter interval are presented in the section Sensitivity Analyses. While using a longer serial interval has very little impact on the weekly forecasts produced, it results in much higher estimates of transmissibility.

Figure 1 Serial Interval distributions used in the analysis. Here the serial interval relates to death and characterise the time between the deaths of an infector their infectee. Our main analysis assumes a gamma distribution with a mean of 6.48 days and a standard deviation of 3.83 days (shown in green). The shorter serial interval, used for sensitivity analysis, with a mean of 4.80 days and a standard deviation of 2.70 days is shown in purple.

Projections and \(R_t\) estimates

Ensemble Model

Methods

This is an unweighted ensemble of Models 1, 2 and 3. We obtained posterior distribution for all estimated reproduction numbers and forecasted deaths by simply combining the posterior distributions of each model.

Ensemble models, even if built with a relatively simple approach such as adopted here, have been shown to typically perform better than individual models in the context of epidemiology of infectious diseases (6).

Projections

Current and past forecasts

Caution note: We note that in France, a large increase in deaths was reported towards the end of the week starting 30-03-2020. This is largely due to back-reporting of deaths outside hospital settings, and therefore, this is likely to have inflated the estimates \(R_t\). The forecasts of deaths for the coming week are thefore likely to be over-estimated.

Figure 2: Reported daily deaths, current and past forecasts based on the ensemble model. For each country with active transmission (see methods), we plot the observed incidence of deaths (black dots). Past forecasts, where available, are shown in green (median and 95% CrI), while latest forecasts are shown in red (median and 95% CrI). Vertical dashed lines show the start and end of each week (Monday to Sunday).

Estimates of the current Effective Reproduction Number

Figure 3: Latest estimates of effective reproduction numbers by country (median and 95% CrI). We present the estimates of current transmissibility estimated from each method as well as the ensemble estimates.

Summary of results

Table 1: Observed (where available) and forecasted weekly death counts and the estimated levels of transmissibility from the ensemble model for each country with active transmission (see Methods) and for each period for which forecasts were produced. The number of deaths has been rounded to 3 significant figures.

Model 1

Methods

The approach, similar to model 2, was to estimate the current reproduction number (the average number of secondary cases generated by a typical infected individual, \(R_t\)) and to use that to forecast future incidence of death. The current reproduction number was estimated assuming constant transmissibility during a chosen time-window (i.e. one week).

Estimating current transmissibility

Here we relied on a well-established and simple method (7) that assumed the daily incidence, I_t (here representing deaths), could be approximated with a Poisson process following the renewal equation (8):

\[I_t \sim Pois\left( R_t \sum_{s=0}^tI_{t-s}w_s\right)\]

where \(R_t\) is the instantaneous reproduction number and \(w\) is the serial interval distribution. From this a likelihood of the data given a set of model parameters can be calculated, as well the posterior distribution of \(R_t\) given previous observations of incidence and knowledge of the serial interval (9).

We used this approach to estimate \(R_t\) over three alternative time-windows defined by assuming a constant \(R_t\) for either the 2, 3 or 4 weeks prior to the most recent data-point. We made no assumptions regarding the epidemiological situation and transmissibility prior to each time-window. Therefore, no data prior to the time-window were used to estimate \(R_t\), and instead we jointly estimated \(R_t\) as well as back-calculated the incidence before the time-window. Specifically, we jointly estimated the \(R_t\) and the incidence level 100 days before the time-widow. Past incidence was then calculated using the known relationship between the serial interval, growth rate and reproduction number. The joint posterior distribution of \(R_t\) and the early epidemic curve (from which forecasts will be generated) were inferred using Markov Chain Monte Carlo (MCMC) sampling.

The model has the advantage of being robust to changes in reporting before the time-window used for inference.

Forward projections

We used the renewal equation (8) to project the incidence forward, given a back-calculated early incidence curve, an estimated reproduction number, and the observed incidence over the calibration period. We sampled sets of back-calculated early incidence curves and reproduction numbers from the posterior distribution obtained in the estimation process. For each of these sets, we simulated stochastic realisations of the renewal equation from the end of the calibration period leading to projected incidence trajectories.

Projections were made on a 7-day horizon. The transmissibility is assumed to remain constant over this time period. If transmissibility were to decrease as a result of control interventions and/or changes in behaviour over this time period, we would predict fewer deaths; similarly, if transmissibility were to increase over this time period, we would predict more deaths We limited our projection to 7 days only as assuming constant transmissibility over longer time horizons seemed unrealistic in light of the different interventions implemented by different countries and potential voluntary behaviour changes.

Projections

Current and past forecasts

Figure 4: Reported daily deaths, current and past forecasts based on model 1. For each country with active transmission (see Methods), we plot the observed incidence of deaths (black dots). Past forecasts, where available, are shown in green (median and 95% CrI), while latest forecasts are shown in red (median and 95% CrI). Vertical dashed lines show the start and end of each week (Monday to Sunday).

Estimates of the Effective Reproduction Number

Figure 5: Latest estimates of effective reproduction numbers by country (median and 95% CrI). We present the estimates of current transmissibility estimated from model 1.

Summary of results

Table 2: Observed (where available) and forecasted weekly death counts, and estimated levels of transmissibility from Model 1 for each country with active transmission (see methods) and for each period for which forecasts were produced. The number of deaths has been rounded to 3 significant figures.

Model 2

Methods

Estimating current transmissibility

The standard approach to inferring the effective reproduction number at \(t\), \(R_t\), from an incidence curve (with cases at t denoted I_t) is provided by (9). This method assumes that \(R_t\) is constant over a window back in time of size k units (e.g. days or weeks) and uses the part of the incidence curve contained in this window to estimate \(R_t\). However, estimates of \(R_t\) can depend strongly on the width of the time-window used for estimation. Thus mis-specified time-windows can bias our inference. In (10) we use information theory to extend the approach of Cori et al. to optimise the choice of the time-window and refine estimates of \(R_t\). Specifically:

We integrate over the entire posterior distribution of \(R_t\), to obtain the posterior predictive distribution of incidence at time t+1 as P(I_t+1 | I₁^t) with I₁^t as the incidence curve up to t. For a gamma posterior distribution over \(R_t\) this is analytic and negative binomial ((10) for exact formulae).
We compute this distribution sequentially and causally across the existing incidence curve and then evaluate every observed case-count according to this posterior predictive distribution. For example at t = 5, we pick the true incidence value I₅* and evaluate the probability of seeing this value under the predictive distribution i.e. P(I₅ = I₅* | I₁⁴).

This allows us to construct the accumulated predictive error (APE) under some window length k and under a given generation time distribution as:

\[\text{AP}E_{k} = \sum_{t = 0}^{T - 1}{- \log{P\left( I_{t + 1} = I_{t + 1}^{*}\ \right|\ I_{t - k + 1}^{t})\ \ }}\]

The optimal window length k* is then \(k^{*} = \arg{\min_{k}{\text{AP}E_{k}}}\). Here T is the last time point in the existing incidence curve.

Forward Projections

Forward projections are made assuming that the transmissibility remains unchanged over the projection horizon and same as the transmissibility in the last time-window. The projections are made using the standard branching process model using a Poisson offspring distribution.

Projections

Current and past forecasts

Figure 6: Reported daily deaths, current and past forecasts based on model 2. For each country with active transmission (see Methods), we plot the observed incidence of deaths (black dots). Past forecasts, where available, are shown in green (median and 95% CrI), while latest forecasts are shown in red (median and 95% CrI). Vertical dashed lines show the start and end of each week (Monday to Sunday).

Effective Reproduction Number Estimates

Figure 7: Latest estimates of effective reproduction numbers by country (median and 95% CrI). We present the estimates of current transmissibility from model 2.

Summary of results

Table 3: Observed (where available) and forecasted weekly death counts and the estimated levels of transmissibility from Model 2 for each country with active transmission (see methods) and for each period for which forecasts were produced. The number of deaths has been rounded to 3 significant figures.

Model 3

Methods

The methods for this model are presented in detail in the section “Case Ascertainment”. Please note that for this model, we do no estimate the effective reproduction number to forecast ahead.

Projections

Current and Past Forecasts

Figure 8: Reported daily deaths, current and past forecasts based on model 3. For each country with active transmission (see Methods), we plot the observed incidence of deaths (black dots). Past forecasts, where available, are shown in green (median and 95% CrI), while latest forecasts are shown in red (median and 95% CrI). Vertical dashed lines show the start and end of each week (Monday to Sunday).

Summary of results

Sensitivity Analyses

Results below use a shorter serial interval distribution (mean 4.80 days, SD 2.70 days). Only results from the ensemble model are presented here.

Projections

Figure 9: Reported daily deaths, current and past forecasts based on the model ensemble. For each country with active transmission (see methods), we plot the observed incidence of deaths (black dots). Past forecasts, where available, are shown in green (median and 95% CrI), while latest forecasts are shown in red (median and 95% CrI). Vertical dashed lines show the start and end of each week (Monday to Sunday).

Effective Reproduction Number Estimates

Figure 10: Latest estimates of effective reproduction numbers by country (median and 95% CrI). We present the estimates of current transmissibility estimated from each method as well as the ensemble estimates.

Case Ascertainment

Methods

Objectives

Estimate trends in case ascertainment and the ratio of deaths to reported cases.
Use these to estimate the true size of the epidemic.
Use these to forecast the number of deaths in the coming week.

Assumptions

We assume

that deaths due to COVID-19 are perfectly reported;
a known distribution for delay from report to death (gamma distribution with mean 10 days and standard deviation 2 days); and,
a known distribution for CFR (2).

Let \(D_{i, t}\) be the number of deaths in location \(i\) at time \(t\). Let \(I_{i, t}^r\) be the reported number of cases in location \(i\) at time \(t\) and \(I_{i, t}^{true}\) be the true number of cases. We assume that the reporting to death delay \(\delta\) is distributed according to a Gamma distribution with mean \(\mu\) and standard deviation \(\sigma\). That is,

\[\delta \sim \Gamma(\mu, \sigma).\]

Let \(r_{i, t}\) be the ratio of deaths to reported cases in location \(i\) at time \(t\). We assume that deaths are distributed according to a Binomial distribution thus: \[ D_{i, t} \sim Binom\left( \int\limits_0^{\infty}{\Gamma(x \mid \mu, \sigma)I_{i, t - x}^{r}dx} , r_{i, \mu}\right). \]

This allows us to obtain a posterior distribution for \(r_{i, t}\).

Case ascertainment is defined as:

\[ \rho_{i, t} = \frac{CFR}{r_{i, t}}.\]

Thus a posterior distribution for \(\rho_{i, t}\) can be obtained using the posterior distribution for \(r_{i, t}\) and the posterior distribution for CFR.

Combining the CFR and the case ascertainment, we can estimate the true number of cases in the epidemic at any point.

For the period over which we have information on deaths i.e., up to time \(t - \mu\), we use the posterior distribution of CFR to obtain \(I_{i, t}^{true}\). The true number of cases in a location \(i\) at time \(t\) is the sum cases that did not die and the number of deaths. \[ I_{i, t}^{true} \sim D_{i, t - \mu} + NBin(D_{i, t - \mu}, CFR). \]

In this formulation, the negative binomial is parameterised as \(NBin(n, p)\) where \(n\) is the number of failure (i.e. death), and \(p\) is the probability of observing a failure.

For the period over which we do not have information on deaths i.e., after time \(t - \mu\), we use the posterior distribution of case ascertainment to obtain

\[ I_{i, t}^{true} \sim I_{i, t}^{r} + NBin(I_{i, t}^{r}, \rho_{i, t}). \]

To obtain forecast of deaths, we rely on reported cases to obtain \(\int\limits_0^{\infty}{\Gamma(x \mid \mu,\sigma)I_{i, t - x}^{r}dx}\). As cases reported in the coming week may die within the same week ( i.e. for \(x \in \{0,7\}\), \(\Gamma(x \mid \mu,\sigma) > 0\)), we estimate new reporting cases in the coming week by sampling from a Gamma distribution with mean and standard deviation estimated from the number of observed cases in the last week.

While this assumes no growth or decline in the coming week, this baseline assumtpion is justfiable as a null-hypthesis scenario as it does not influence our results dramatically given the contribution to deaths due to those being very small (i.e. less than 2%).

We therefore obtain the forecasted number of deaths as: \[ D_{i, t} \sim Binom\left( \int\limits_0^{\infty}{\Gamma(x \mid \mu, \sigma)I_{i, t - x}^{r}dx} , r_{i, \mu}\right). \]

where \(r_{i, \mu}\) is the estimated ratio of deaths to reported cases for the last week of data, and \(\int\limits_0^{\infty}{\Gamma(x \mid \mu,\sigma)I_{i, t - x}^{r}dx}\) relies on observed reported cases up to the last day with available and estimated reported cases as described above.

Temporal trend in the ratio of deaths to reported cases

Starting in March, we compute the average and 95% CI for the ratio of deaths to reported cases (with a 10-day delay) using a moving window of 7 days.

Figure 11: Temporal trends in the ratio of reported deaths to reported cases 10 days prior (medians and 95% CIs, solid lines and bands respectively). Also plotted are the reported deaths (red dots) and reported cases (black dots). The reported number of deaths and cases have been re-scaled so that the maximum recorded numbers of deaths or cases (with a 10-day delay) reaches 1.

Note that if deaths exceed the number of reported cases 10 days before, we set the ratio at 1 (95% CI 1-1)

Estimating ascertainment

Table 4: Results summary for reporting and case ascertainment. We present (i) the estimated ratio of deaths to reported cases 10 days prior per country (mean and 95% CI) during the last 14 days; (ii) the estimated case ascertainment per country (mean and 95% CI) during the last 14 days; (iii) the estimated factor to be multiplied by observed number of cases to obtain the true number of cases (scaling factor x observed cases). Mean and 95% CI, estimated for the last 14 days; (iv) the observed and predicted number of cases in the last week (week starting 22-03-2020). Estimates for the number of deaths have been rounded to 3 significant figures. Estimated case ascertainment, factors to real size and predicted true number of cases assume perfect reporting of deaths.

Authors

Sangeeta Bhatia, Anne Cori, Kris V Parag, Kylie E C Ainslie, Marc Baguelin, Samir Bhatt, Adhiratha Boonyasiri, Olivia Boyd, Lorenzo Cattarino, Zulma Cucunubá, Gina Cuomo-Dannenburg, Amy Dighe, Ilaria Dorigatti, Sabine van Elsland, Rich FitzJohn, Han Fu, Katy Gaythorpe, Will Green, Arran Hamlet, David Haw, Sarah Hayes, Wes Hinsley, Natsuko Imai, David Jorgensen, Edward Knock, Daniel Laydon, Swapnil Mishra, Gemma Nedjati-Gilani, Lucy C Okell, Steven Riley, Hayley Thompson, Juliette Unwin, Robert Verity, Michaela Vollmer, Caroline Walters, Hao Wei Wang, Patrick GT Walker, Oliver Watson, Charles Whittaker, Yuanrong Wang, Peter Winskill, Xiaoyue Xi, Azra C Ghani, Christl A Donnelly, Neil M Ferguson, Pierre Nouvellet

References

The forecasts produced use the reported daily counts of deaths per country available on the ECDC website:

https://www.ecdc.europa.eu/en/publications-data/download-todays-data-geographic-distribution-covid-19-cases-worldwide

1. Coronavirus disease 2019 (COVID-19) Situation Report – 76. 2020

2. Verity R, Okell L, Dorigatti I, Winskill P, Whittaker C, et al. 2020. Estimates of the severity of coronavirus disease 2019: A model-based analysis. Lancet Infectious Diseases

3. Deaths relating to the coronavirus (COVID-19). 2020

4. Wang Y, Teunis PF. 2020. Strongly heterogeneous transmission of covid-19 in mainland china: Local and regional variation. medRxiv

5. Ferguson N, Laydon D, Nedjati Gilani G, Imai N, Ainslie K, et al. 2020. Report 9: Impact of non-pharmaceutical interventions (npis) to reduce covid19 mortality and healthcare demand

6. Viboud C, Sun K, Gaffey R, Ajelli M, Fumanelli L, et al. 2018. The RAPIDD Ebola forecasting challenge: Synthesis and lessons learnt. Epidemics. 22:13–21

7. Nouvellet P, Cori A, Garske T, Blake IM, Dorigatti I, et al. 2018. A simple approach to measure transmissibility and forecast incidence. Epidemics. 22:29–35

8. Fraser C. 2007. Estimating individual and household reproduction numbers in an emerging epidemic. PloS One. 2(8):

9. Cori A, Ferguson NM, Fraser C, Cauchemez S. 2013. A new framework and software to estimate time-varying reproduction numbers during epidemics. American journal of epidemiology. 178(9):1505–12

10. Parag KV, Donnelly CA. 2019. Optimising renewal models for real-time epidemic prediction and estimation. bioRxiv, p. 835181

Short-term forecasts of COVID-19 deaths in multiple countries

April 08, 2020

Introduction

Executive Summary

Detailed Summary

Objectives and caveats

Key results on transmissibility and forecasting

Key results on case ascertainment

Methods

Projections and \(R_t\) estimates

Ensemble Model

Methods

Projections

Estimates of the current Effective Reproduction Number

Summary of results

Model 1

Methods

Projections

Estimates of the Effective Reproduction Number

Summary of results

Model 2

Methods

Projections

Effective Reproduction Number Estimates

Summary of results

Model 3

Methods

Projections

Summary of results

Sensitivity Analyses

Projections

Effective Reproduction Number Estimates

Case Ascertainment

Methods

Temporal trend in the ratio of deaths to reported cases

Estimating ascertainment

Authors

References